Why is the Flash logo popping up and will not go away?

Flash Player was discontinued on Dec 31 2020, at the end of a three year pre-communication period.

https://theblog.adobe.com/adobe-flash-update/

As of January 12, Flash Player will no longer load content, unless that content is explicitly allowed through specific configuration directives in the Flash Player config file.  The icon you're seeing is a product of this.  Something is launching Flash Player, and we're behaving as designed.  We don't load anything, and display that placeholder icon.

I don't work on Lightroom, so I can't speak to what's happening there.  I assume they're loading a page or modal dialog with Flash content in it.  I'm also sure that more recent versions of Lightroom address the root-cause of the issue, which is that dependencies on Flash Player need to be retired at this point. 

It's possible that Lightroom experts in the Lightroom support community could offer you advice on a workaround.  There are literally hundreds of downstream Adobe products that have leveraged Flash in various ways over the last ~15 years.  We're sort of like a hammer factory.  I can tell you about the hammer, but I don't have much authoritative insight about the things that people ultimately used that hammer to build.

We strongly recommend that you uninstall Flash Player at this point.  For your convenience, I've linked the relevant uninstaller packages below. 

Uninstall Flash Player - Windows:
https://helpx.adobe.com/flash-player/kb/uninstall-flash-player-windows.html

Uninstall Flash Player - Mac:
https://helpx.adobe.com/flash-player/kb/uninstall-flash-player-mac-os.html

Microsoft Update for Removal of Flash Player for IE and Edge on Windows 8 and higher

https://support.microsoft.com/en-us/help/4577586/update-for-removal-of-adobe-flash-player 

Hope that helps!